Mathematical Problems in Engineering

Volume 2015, Article ID 510738, 12 pages

http://dx.doi.org/10.1155/2015/510738

## Robust Adaptive PID Controller for a Class of Uncertain Nonlinear Systems: An Application for Speed Tracking Control of an SI Engine

^{1}Department of Industrial Engineering, Khon Kaen University, Khon Kaen 40002, Thailand^{2}Department of Mechanical Engineering, Khon Kaen University, Khon Kaen 40002, Thailand

Received 12 October 2014; Accepted 25 February 2015

Academic Editor: Kacem Chehdi

Copyright © 2015 Tossaporn Chamsai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The sliding mode control (SMC) technique with a first-order low-pass filter (LPF) is incorporated with a new adaptive PID controller. It is proposed for tracking control of an uncertain nonlinear system. In the proposed control scheme, the adaptation law is able to update the PID controller online during the control process within a short period. The chattering phenomenon of the SMC can be alleviated by incorporation of a first-order LPF, while the robustness of the control system is similar to that of the sliding mode. In the closed-loop control analysis, the convergence condition in the reaching phase and the existence condition of the sliding mode were analyzed. The stability of the closed-loop control is guaranteed in the sense of Lyapunov’s direct method. The simulations and experimental applications of a speed tracking control of a spark ignition (SI) engine via electronic throttle valve control architecture are provided to verify the effectiveness and the feasibility of the proposed control scheme.

#### 1. Introduction

The presence of uncertain nonlinearity in physical systems has been extensively studied because most of the real systems are rather complex dynamical nonlinear with large uncertainties which can affect inaccuracy and poor robustness of the control system. To do this, numerous control schemes have been developed; for example, sliding mode control [1–3], intelligent control [4, 5], feedback linearization [6], and adaptive control [7–9].

On the other hand, due to its simplicity in architecture and simple design, the proportional-integral-derivative (PID) control is acceptable and has been extensively applied in many practical applications. The key for designing a high capability of PID controller depends on the determination of the PID gain parameters which should be properly adjusted, and this has led to the developments for self-tuning methods of the three parameters of the PID controller. Recently, adaptive control techniques are generally applied for online-tuning of the PID controller. The flexibility of the PID tuning lies in the determination of the basic PID gains that can be adjusted online according to adaptation laws and using the error signal in order to realize the online adjustable gains during a control procedure [10–13]. However, the adaptive control strategy is capable of handling only constant parametric uncertainty, inadequate robustness for against external disturbance, and an accurate system model that is required [14].

The sliding mode control (SMC) method is one of the control strategies to dominate the parametric uncertainties and external disturbances, while the control principle is without precise system model information [15–18]. Nevertheless, if the sliding mode exists, the chattering phenomenon is the main obstacle for SMC application [19–21]. In much research work, alleviations chattering phenomenon of the sliding mode incorporated with a low-pass filter has been studied [22–26] because it can make a compromise between the alleviation of the chattering and the control accuracy. Incorporation of the SMC with closed-loop filtering is capable to realize the acquisition of control signal and the approximation of disturbances [24], while the robustness is similar to that of the sliding mode.

In this study, the SMC with a first-order LPF is incorporated with a new adaptive PID controller. It is proposed for perfect tracking control tasks of uncertain nonlinear systems. In the proposed control scheme, the PID controller is adjusted during the control procedure according to the adaptation laws, while the chattering of the control signal can be alleviated by the first-order LPF. The stability of the closed-loop control can be guaranteed in the sense of Lyapunov’s direct method [27, 28]. The effectiveness and the feasibility of the proposed control scheme are assessed in the problem of a speed tracking control of a spark ignition (SI) engine via electronic throttle valve control architecture. Interest is due to the delay of the intake manifold filling dynamic and the induction-power delay is the drawback in practice for engine speed control through the throttle valve regulation method [29, 30]. Furthermore, the engine system is a rather complex mechanism, multiactuation, and largely uncertain nonlinearity phenomena which are present in the engine mechanism [31]. Therefore, the engine speed control is a well-known challenge in the problem of uncertain nonlinear control systems [32–34].

Increasing transient performance and tracking accuracy of speed responses are usefulness during acceleration of the vehicle at any operating condition, especially in the transition mode of the hybrid operating system of hybrid vehicles [35, 36]. The goal of the engine speed control resulting from the speed response is to be able to track a desired speed at any operating condition [37], especially at the transient-state that has significant effects on optimizing the maneuverability of an engine speed control. High accuracy of speed tracking control leads to the achievement of an optimal engine operating point for other applications. In addition, increased engine performance, reduced fuel consumption, and exhaust emission are other benefits of an optimized engine speed control [38, 39].

The remainder of this paper is organized as follows. Firstly, we present the controller design and closed-loop control analysis. In Section 3, a description of the engine speed control model is presented. In Section 4, the simulations and experimental results are presented to verify the effectiveness of the development control approach. Finally, the research conclusions are presented.

#### 2. Controller Design and Closed-Loop Control Analysis

##### 2.1. System Formulation

In an engine system, the two first-order dynamic elements which are rotational dynamics and the manifold filling dynamics behave as a second-order system [40]. However, the system order can be reduced due to substitution of the engine torque as described by the mean-value method [41] into the first-order crankshaft rotational dynamics from Newton’s law. Thus, for simplicity, this work realizes the system to be a first-order uncertain nonlinear system, satisfying uncoupling and matching conditions [28, 40]. It can be described in a canonical form aswhere is the first-order of the state vector, is time, is the known nonlinear function, is the global state vector for the nonlinear square system, is the number of independent coordinates of , , is the control input, is the uncertain element that presents only in the highest order of the system, and , , is the control gain distribution matrix which is positive definite in all arguments. The aim is to force the system state to reach the desired state so that the error , . Not only does a control law have to steer the response to the desired value, but it must have the ability to overcome a system’s uncertainties also.

In the work of Xu et al. [24], uncertainties can be estimated by adding a second low-pass filter; as a result, the switching gain is reduced to the minimum level, while the first low-pass filter smooths out the switching control. However, during the filter’s reaching phase, there are no control inputs from both low-pass filters and a feed forward term from an equivalent control may not have the capability to override the uncertainties. If another control input, during the reaching phase, can be adjusted suitably, better transient control can be achieved. The assumptions below are made for control law derivation in the next section.

*Assumption 1. *The functions , , and are continuous in for all and continuously differentiable.

*Assumption 2. *The uncertainties are within the range space of the control distribution matrix and uncertainty variation does not affect the control direction.

*Assumption 3. *The estimated control distribution matrix is invertible and continuously dependent on parametric uncertainty [28].

*Assumption 4. *Let be upper bounded by and upper bounded by , where and are known constants.

*Assumption 5. *The input matrix is bounded by and its derivative is upper bounded by , where , , and are known constants.

##### 2.2. Closed-Loop Control Design

To design the control law, a definition of the sliding function similar to the work of Slotine and Li [28] is adapted to the first-order system. The sliding function can be made on each state as

Then, the -dimensional sliding function and its first derivative for the system in (1) can be expressed aswhere has components satisfying the sliding function stated earlier and is a positive constant coefficient matrix with full row rank. Its argument is a gradient vector of the sliding function [42] which is unity in each sliding surface due to the first-order system.

From Figure 1, the control input is where is the low-frequency pass filter signal which is a result of a first-order low-pass filter of the switching signal ; . Let , , and denote the control signal of the adaptive PID tuning. is an approximation of the control input that neglects the last term of the RHS of (1) and is the unit matrix. Therefore, an approximation can be obtained similar to the idea of the equivalent control input [15] which is